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The Classification Problem

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PGM: Tirgul 11Na?ve Bayesian Classifier +Tree Augmented Na?ve Bayes(adapted from tutorial by Nir Friedman and Moises Goldszmidt

Age

Sex

ChestPain

RestBP

Cholesterol

BloodSugar

ECG

MaxHeartRt

Angina

OldPeak

Heart Disease

- From a data set describing objects by vectors of features and a class
- Find a function F: featuresclass to classify a new object

Vector1= <49, 0, 2, 134, 271, 0, 0, 162, 0, 0, 2, 0, 3> Presence

Vector2= <42, 1, 3, 130, 180, 0, 0, 150, 0, 0, 1, 0, 3>Presence

Vector3= <39, 0, 3, 94, 199, 0, 0, 179, 0, 0, 1, 0, 3 >Presence

Vector4= <41, 1, 2, 135, 203, 0, 0, 132, 0, 0, 2, 0, 6 >Absence

Vector5= <56, 1, 3, 130, 256, 1, 2, 142, 1, 0.6, 2, 1, 6 >Absence

Vector6= <70, 1, 2, 156, 245, 0, 2, 143, 0, 0, 1, 0, 3 >Presence

Vector7= <56, 1, 4, 132, 184, 0, 2, 105, 1, 2.1, 2, 1, 6 >Absence

- Predicting heart disease
- Features: cholesterol, chest pain, angina, age, etc.
- Class: {present, absent}

- Finding lemons in cars
- Features: make, brand, miles per gallon, acceleration,etc.
- Class: {normal, lemon}

- Digit recognition
- Features: matrix of pixel descriptors
- Class: {1, 2, 3, 4, 5, 6, 7, 8, 9, 0}

- Speech recognition
- Features: Signal characteristics, language model
- Class: {pause/hesitation, retraction}

- Memory based
- Define a distance between samples
- Nearest neighbor, support vector machines

- Decision surface
- Find best partition of the space
- CART, decision trees

- Generative models
- Induce a model and impose a decision rule
- Bayesian networks

- Bayesian classifiers
- Induce a probability describing the data
P(A1,…,An,C)

- Impose a decision rule. Given a new object < a1,…,an >
c = argmaxC P(C = c | a1,…,an)

- Induce a probability describing the data
- We have shifted the problem to learning P(A1,…,An,C)
- We are learning how to do this efficiently: learn a Bayesian network representation for P(A1,…,An,C)

- Let ci be the true class, and let lj be the class returned by the classifier.
- A decision by the classifier is correct if ci=lj, and in error if ci lj.
- The error incurred by choose label lj is
- Thus, had we had access to P, we minimize error rate by choosing liwhenwhich is the decision rule for the Bayesian classifier

- Output: Rank over the outcomes---likelihood of present vs. absent
- Explanation: What is the profile of a “typical” person with a heart disease
- Missing values: both in training and testing
- Value of information: If the person has high cholesterol and blood sugar, which other test should be conducted?
- Validation: confidence measures over the model and its parameters
- Background knowledge: priors and structure

Partition the data set in n segments

Do n times

Train the classifier with the green segments

Test accuracy on the red segments

Compute statistics on the n runs

Variance

Mean accuracy

Accuracy: on test data of size m

Acc =

D1

D2

D3

Dn

Run 1

Run 2

Run 3

Run n

Original data set

Outcome

Age

MaxHeartRate

Vessels

STSlope

Angina

BloodSugar

OldPeak

ChestPain

RestBP

ECG

Thal

Sex

Cholesterol

- Efficiency in learning and query answering
- Combine knowledge engineering and statistical induction
- Algorithms for decision making, value of information, diagnosis and repair

Heart disease

Accuracy = 85%

Data source

UCI repository

Problems with BNs as classifiers

When evaluating a Bayesian network, we examine the likelyhood of the model B given the data D and try to maximize it:

When Learning structure we also add penalty for structure complexity and seek a balance between the two terms (MDL or variant). The following properties follow:

- A Bayesian network minimized the error over all the variables in the domain and not necessarily the local error of the class given the attributes (OK with enough data).
- Because of the penalty, a Bayesian network in effect looks at a small subset of the variables that effect a given node (it’s Markov blanket)

Problems with BNs as classifiers (cont.)

Let’s look closely at the likelyhood term:

- The first term estimates just what we want: the probability of the class given the attributes. The second term estimates the joint probability of the attributes.
- When there are many attributes, the second term starts to dominate (value of log is increased for small values).
- Why not use the just the first term? We can no longer factorize and calculations become much harder.

C

insulin

F1

F2

F3

F4

F5

F6

age

mass

glucose

pregnant

dpf

Diabetes in

Pima Indians

(from UCI repository)

- Fixed structure encoding the assumption that features are independent of each other given the class.
- Learning amounts to estimating the parameters for each P(Fi|C) for each Fi.

The Naïve Bayesian Classifier (cont.)

What do we gain?

- We ensure that in the learned network, the probability P(C|A1…An) will take every attribute into account.
- We will show polynomial time algorithm for learning the network.
- Estimates are robust consisting of low order statistics requiring few instances
- Has proven to be a powerful classifier often exceeding unrestricted Bayesian networks.

C

F1

F2

F3

F4

F5

F6

- Common practice is to estimate
- These estimate are identical to MLE for multinomials

- Naïve Bayes encodes assumptions of independence that may be unreasonable:
Are pregnancy and age independent given diabetes?

Problem: same evidence may be incorporated multiple times (a rare Glucose level and a rare Insulin level over penalize the class variable)

- The success of naïve Bayes is attributed to
- Robust estimation
- Decision may be correct even if probabilities are inaccurate

- Idea: improve on naïve Bayes by weakening the independence assumptions
Bayesian networks provide the appropriate mathematical language for this task

C

mass

dpf

pregnant

age

glucose

F1

F2

F4

F5

F6

insulin

F3

- Approximate the dependence among features with a tree Bayes net
- Tree induction algorithm
- Optimality: maximum likelihood tree
- Efficiency: polynomial algorithm

- Robust parameter estimation

Optimal Tree construction algorithm

The procedure of Chow and Lui construct a tree structure BT that maximizes LL(BT |D)

- Compute the mutual information between every pair of attributes:
- Build a complete undirected graph in which the vertices are the attributes and each edge is annotated with the corresponding mutual information as weight.
- Build a maximum weighted spanning tree of this graph.
Complexity: O(n2N) + O(n2) + O(n2logn) = O(n2N) where n is the number of attributes and N is the sample size

Tree construction algorithm (cont.)

It is easy to “plant” the optimal tree in the TAN by revising the algorithm to use a revised conditional measure which takes the conditional probability on the class into account:

This measures the gain in the log-likelyhood of adding Ai as a parent of Aj when C is already a parent.

Problem with TAN

When evaluating parameters we estimate the conditional probability P(Ai|Parents(Ai)). This is done by partitionaing the data according to possible values of Parents(Ai).

- When a partition contains just a few instances we get an unreliable estimate
- In Naive Bayes the partition was only on the values of the classifier (and we have to assume that is adequate)
- In TAN we have twice the number of partitions and get unreliable estimates, especially for small data sets.
Solution:

where s is the smoothing bias and typically small.

100

- 25 Data sets from UCI repository
- Medical
- Signal processing
- Financial
- Games

- Accuracy based on 5-fold cross-validation
- No parameter tuning

95

90

85

Naïve Bayes

80

75

70

65

65

70

75

80

85

90

95

100

TAN

- 25 Data sets from UCI repository
- Medical
- Signal processing
- Financial
- Games

- Accuracy based on 5-fold cross-validation
- No parameter tuning

100

95

90

85

C4.5

80

75

70

65

65

70

75

80

85

90

95

100

TAN

- Can we do better by learning a more flexible structure?
- Experiment: learn a Bayesian network without restrictions on the structure

100

95

90

85

80

75

70

65

65

70

75

80

85

90

95

100

- 25 Data sets from UCI repository
- Medical
- Signal processing
- Financial
- Games

- Accuracy based on 5-fold cross-validation
- No parameter tuning

Bayesian Networks

TAN

- Bayesian networks provide a useful language to improve Bayesian classifiers
- Lesson: we need to be aware of the task at hand, the amount of training data vs dimensionality of the problem, etc

- Additional benefits
- Missing values
- Compute the tradeoffs involved in finding out feature values
- Compute misclassification costs

- Recent progress:
- Combine generative probabilistic models, such as Bayesian networks, with decision surface approaches such as Support Vector Machines